On superior limits for the increments of Gaussian processes

STATISTICS& PROBABILITY LETTERS ELSEVIER

Statistics & Probability Letters 35 (1997) 289-296

On superior limits for the increments of Gaussian processes 1 Kyo Shin Hwang a, Yong Kab Choi a,,, Jong Soo Jung b a Department of Mathematics, Gyeongsan O National University, Chinju 660-701, South Korea b Department of Mathematics, Dong-A University, Pusan 604-714, South Korea Received 1 May 1996

Abstract Let {X(t);t>~O} be a Gaussian process with stationary increments E { X ( t + s ) - X(t)} 2 = a~(s). Let at(t>~O) be a nondecreasing function of t with 0 < a, ~~l. On the contrary, if l i m i n f k ~ { ( t k + l

--tk)/atk}>~ 1, then we have a value 6 almost surely, where 6 = i n f { 7 > 0 ; ~-~k(tk(log(,)tk)/atk)-~2~ 1. Also in the case of stationary Gaussian sequence, we obtain the similar results. (~) 1997 Elsevier Science B.V. A M S 1991 classification: 60F05, 60G05 Keywords: Wiener process; Gaussian process; Law of large numbers; Law of iterated logarithm; Regularly varying function and Borel-Cantelli lemma

I. Introdution Let { W ( t ) ; O < ~ t < c c } be a standard W i e n e r process defined on a probability space ( f 2 , ~ , P ) . Let at ( t > 0 ) be a n o n d e c r e a s i n g function o f t with 0 ~ l } be an increasing sequence diverging to c~. (i) I f lim s u p k ~ ( t k + l -- tk)/atk < 1, then lim sup flt~

k---+oc~

IW(t~ + atk) -- W(tk)l

-----lim sup

sup

k---+c~ O~s~atk

fltk [W(tk + s) - W(tk)] = 1

a.s.

(1)

where flt~ = {2at~(log(tk/atk ) ÷ log log tk)}-1/2.

* Corresponding author. Tel.: +82 591 751 5965; fax: +82 591 751 5965; e-mail: [email protected]. 1 The authors were supported in part by the Basic Science Research Institute program, Ministry of Education, Korea, 1996, Project No. BSRI-96-1405. 0167-7152/97/$17.00 (~) 1997 Elsevier Science B.V. All rights reserved PH S01 67-71 52(97 )0002 5-4

K.S. H w a n 9 et al. I Statistics & Probability Letters 35 (1997) 2 8 9 - 2 9 6

290

(ii) If lim infk--,~ (tk+l - tk)/atk > 1, then limsup/~tk IW(tk +atk)-- W(tk)l = limsup k---~c~

sup

t~t~ IW(tk + s ) -

W(tk)l =~*

a.s.

(2)

k---~oo O<~s<~atk

where ~*=inf{ 7>0;Z(9(tk))-~2<°°}k

and

9(t~)=tk(logtk)/atk.

In this paper we are going to extend the above results to Gaussian processes. Let {X(t):O<<.t0. Further we assume that a(t), t > 0 , is a nondecreasing continuous concave, regularly varying function at exponent ~ (0 < ~ < 1) at infinity (e.g., if {X(t):O<<.t 0 ) be a nondecreasing function of t with 0
f(t)-

t 10g(n) t

,

n>~l,

at

f ( t ) is an increasing function of t, and log(i ) t = log(log(i_l)t), i~>2, log0)t = log t. Define continuous parameter processes Gl(t) and Gz(t) by Gl(t)=

sup I X ( t + s ) - X ( t ) l O<.s<~at

and G2(t) = [X(t + at) - X ( t ) l . 2. Main results

Theorem 1. Let {X(t); 0 ~ O) be a nondecreasing function of t with 0 < a t <<.t. I f {tk; k = 1,2 .... } is an increasin9 sequence diverging to oo such that tk+ 1 -- t k

lim sup - k---+oo

< 1,

(.)

ark

then limsup?tkGi(tk)= 1,

a.s., i = 1,2,

(3)

k---+ oo

where ?tk = { 2aZ (atk ) log f ( tk ) } -1/2. We note that ?tk >~fltk for large k in case of the Wiener process. It is interesting to compare (1) and (3). Example 1. (i) For k~> 1, let tk = k n for some r/~> 1, and ark = ( ( k + 1)/(k+2))~tk. Then the condition (*) is satisfied and hence we have (3).

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(ii) If we take tk >t e k, then the condition ( , ) is not satisfied because we cannot choose at such as at = pt for 0 < p < ~ l .

Theorem 2.

Let {X(t); 0 ~ O) be a nondecreasing function o f t with O
lim inf - k---*oo

~> 1,

(**)

atk

then limsupTtkGi(tk)=e**,

a.s., i = 1,2,

k---* cx~

where e ' * * = i n f { 7 > O : Z ("f ( t k ) ) -k? 2 < c ¢ }

E x a m p l e 2. For k~> 1, let tk = exp(exp(k + 1) 2) and at =t/loglogt. Then the condition (**) is satisfied and hence we obtain e** = v/2/2. Moreover, if we take at = 1, then we get e** = 0. We obtain similar results as Theorems 1 and 2 for partial sums of a stationary Gaussian sequence: Let {Xn;n = 1,2 .... } be a stationary Gaussian sequence with X 0 = 0 , EX1 = 0 , EX ( = 1 and E(X1XI+n)<~O for all n n = 1,2 . . . . . We define partial sums S ( n ) = )-'~i=l X/, S(0)---0 and set a2(n)=ES2(n). Assume that tr(n) can be extended to a continuous function G(t) of t > 0 which is nondecreasing and regularly varying with exponent ct ( 0 < ~ < 1) at ~ . Suppose that {an;n = 1,2 .... } is a nondecreasing sequence of positive integers such that 1 ~
f(n)=nl°g(i)n,

i~>1,

an

is an increasing function of n, and define discrete time parameter processes Gl(nk) and G2(nk) by

G1(nk)=

max

IS(nk + j ) -

S(nk)[

1 <<.j<~ank

and G2(nk) = IS(n~ + ank ) -

S(nk)l,

respectively, where { n k ; k = 1,2 .... } is an increasing sequence of positive integers diverging to c~. By the same way as in the proofs of Theorems 1 and 2, we have the following Theorem 3. Under the above statements of {An}, Yn and Gi(nk), i = 1,2, we have the following: (i) I f lim sup nk+l - nk < 1, k-.-*oo

am

K.S. Hwan9 et al. / Stat&tics & Probability Letters 35 (1997) 289-296

292

then limsupTn~ Gi(nk)= 1,

a.s., i = 1,2.

k---+ e~

(ii) I f lim inf nk+l - nk >/1, k---*oo

ank

then limsupT,, Gi(nk)=e **,

a.s., i = 1 , 2 ,

k---*e~

where

= i n f { 7 > 0 " Z ( f ("n k ) ) - ~k2 < ° c }

E x a m p l e 3. Let

{X(t);O<<.t
E { X ( t ) X ( s ) } = ½{It[ = +

Isl

a fractional Brownian motion with the covariance function

- I t -s[~},

0<~<1.

Then E { X ( t ) - S ( s ) } 2 = It - sl ~. Define random variables Xo=0, Xn=X(n)-X(n-1),

n = l , 2 .....

n

S(n) = Z X /

and

S(0)=0.

i=1

Then a2(n ) = ES2(n ) = EX2(n ) = n • and { X n : n = l , 2 .... } is a stationary Gaussian sequence with EX1 = 0 , E X ~ = I and E(X1XI+~)<~O for all n = 1,2 ..... So we have Theorem 3. In particular if ~--1/2, then {X~;n= 1,2 .... } is an i.i.d. Gaussian sequence with EX1 = 0 and EX12 = 1.

3. Proofs

In order to prove Theorems 1 and 2, we need some lemmas: Lemma 1 (Choi, 1991, p. 202). Let {X(t);O<<.t < e c } be as in Theorems 1 and2. Let m be any big number. Then for any small ~i > 0 there exists a positive constant C~, depending on e ~ such that for all u > 0 p ~ sup X ( t + s) - X ( t ) } IO<<.s<~m a(m) > u <~Ce,ue -u2/(2+E').

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Lemma 2 (Slepian, 1962). Let {X(t); t E T} and {Y(t); t E T} be centered Gaussian processes such that ExZ(t) =EY2(t) for all t E T and E(X(t)X(s)} <~E{Y(t)Y(s)} for all s,t E T. Then for any real number u

Proof of Theorem 1. We first prove that

lim sup 7tkGl(tk) ~<1 a.s.

(4)

k--+cx3

For any {tk} with the condition (.), we define an increasing sequence {uk} by

O
and

auk = &+l -- &, k>~l.

For instance, let tk = k ~ for some ~ >~1, =

,k

and

a,k =

) 'k.

Then condition (*) is satisfied, and for all k large, uk 0,

P{Tu, Gl(uk)<~ 1 + e} = P ( °~
>~1 -- 2C~(f (uk ) ) -20+~)2/(2+~') >~ exp(-C'(f(uk)) -1 ) >~ exp(-C'(log(.) uk) -1 )

(5)

where k is big enough and C'> 0 is a constant. By the definition of uk, S := ~

exp(-C'(log(n ) uk) -1 ) = cx~. k

We shall follow the similar proof process as in Vasudeva and Savitha (1993). Set S= Z

exp(-C'(log(.) U2k- 1 )--1) + Z

k

=$1+$2,

exp(-C'(log(.) U2k) -1 ) k

say.

Since {uk} is an increasing sequence, the fact that S = co implies $1 = co =$2. Consider the odd subsequence {t2k-l} of {tk} and define the event Ak = {Tt2k_,G1(t2k- 1) ~<1 + e}. By (5), we have, for all k large,

P(Ak) >1 exp(- C"(log(n) t2k-1 )-1 )

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294

where C " > 0 is a constant. From the fact that u2k-i
P(Ak) >>-e x p ( - C"(log(,) u2k- 1) - 1). Since $1 = oc, we get

Z P(Ak) = 00. k

Also, (6)

t2k-1 "q- at2k_ I <~U2k -F au2k < t2k -F au2k = / 2 k + l .

Setting A~= /I,o~s~<%k_,sup7t2k_,(X(t2k_l+S)-X(t2k_l))~l+t3) and

A'k~= l(O<~s<~a,2k_,sup]~t2k_~(X(t2k_l-}-s)--X(t2k_l))>~ -- l -- e), we have k

k

Let X1 :=

sup

( X ( t 2 k - I + S) - - X ( t 2 k - l ) )

O<~s <~at2k_ I

= X(t2k-i + sl)-- X(t2k-1),

say,

and Yz :=

sup

(X(t2k+l + s) - X(tzk+l))

O~s<~atzk+l = Y(t2k+l +

S2) --

X(t2k+l),

say.

Then, by (6) and the concavity of trz(t), cov(X1,X2) =

E{X(tzk+l + sz)X(t2k-1 + Sl)} - E{X(t2k+l + s2)X(t2k-1 )} - E{X(t2k+l)X(tzk-1 + Sl )} + E{X(t2k+l )X(t2k-1 )}

=

l[{o'2(t2k+l

--

t2k-I +

-- {O'2(t2k+l

--

t2k-1) - a2(t2k+l

~<0.

S2) -- O'2(t2k+l -

--

tZk-1 + S2 - - S l ) }

t2k-1 - - S l ) } ]

K.S. Hwan9 et al. I Statistics & Probability Letters 35 (1997) 289-296

295

Using Lemma 2, we obtain

P(A kI fq A It) <~P(AkI )P(At)I and

p(A~I nat" ).~P(Ak .< ,, )P(At, ) where k ¢ l. It follows from the Borel~antelli lemma that -1 - e ~< lira sup

sup

7t2~_,(X(tzk_l + s ) - X ( t 2 k - 1 ) )

k-.-*~¢~ O<~s<~at2k_l

<~ l + e ,

a.s.

Also, the same result for the even subsequence {t2k} of {tk} is easily obtained. Therefore we have (4). The proof of

lim sup ?t~G2( tk ) >~l

a.s.

k---*c~

is, mutatis mutandis, nearly the same as that of Theorem 1 in Vasudeva and Savitha (1993). Thus the proof of Theorem 1 is complete. Proof of Theorem 2. We first establish that

limsup?tkGl(tk)<~e**

a.s.

(7)

k-*oo

From Lemma 1, we have, for any small e > 0, P{Ttk Gl(tk ) > e ** -}- e}

----P{ 0~sups ~
,X(tk+s)-X(tk).)la(a,

>(e** +e)(21ogf(tk)) 1/2}

<~2C~(f(tk))-z(e** +O2/(2+e) <~ C~,(f(tk)) -(~*'+~'): provided k is big enough, where 0 < e l < e ~ .

From the definition of e**, it follows that

Z P{Ttk Gl(tk )> e** + ~} < c~. k

By appealing to the Borel-Cantelli lemma, we have (7). Next we prove that

limsup?tk(X(tk + atk)-X(tk))>~e** k---~~

Suppose that e** >0. Let q i ( u ) = f +~

Ju

1 e_x2/Zdx,

U~0.

a.s.

(8)

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296

Then, by Femique (1975, p. 71), the inequality 1 e -u2/2 ~<~(u) ~< 4 1 e_U2/2 v / ~ ( u + 1) 3 v / ~ ( u + 1)

holds for all u/> 0. Using the above inequality, one can find positive constants C and K such that, for all

k >~K, P{Yt~ (X(tk + ark ) - X ( t k ) ) >7e** - e} = p ~ X(tk + a r k ) - X(tk) >~(e** - e)(21ogf(tk))l/2~ a( at, ) L J 1 /> ~ { ( ~ * *

- ~)(21ogf(tk)) 1/2 + l } - l ( f ( t k ) ) -(~**-e)2

>~ C(f(t~))-~**-~') 2 where 0 < ~ ~< e < e * * . Set Bk = {~t,(X(tk + ark) -- X ( t k ) ) > ~ ** -- ~}. By the definition of e**, we have

Z P(Bk) = ~ . k>~K The condition (**) implies that there exists K > 0 and the concavity of a2(t), we obtain

such that tk+l >~tk + ark for all k>~K. So, using Lemma 2

P( Bk fq Bt ) <<.P(Bk )P(Bt ) where k # l, and the Borel-Cantelli lemma implies (8). If e** = 0, then Theorem 2 is immediate. Thus the proof of Theorem 2 is complete. []

References Choi, Y.K., 1991. Erd6s-E6nyi type laws applied to Gaussian processes, Doct. Diss., J. Math. Kyoto Univ. 31-1, 191-217. Femique, X., 1975. Evaluations of processus Gaussiens cpmposes, Probability in Banach Spaces, vol. 526, Oberwalbach, Lecture Notes in Mathematics, Springer, Berlin, pp. 67-83. Slepian, D., 1962. The one-sided barrier problem for Gaussian noise, Bell. System Tech. J. 41,463-501. Vasudeva, R., Savitha, S., 1993. On the increments of Wiener process - A look through subsequences, Stochastic Process Appl. 47, 153-158.